To find the number of 6-letter words that begin with T using the letters of the word EQUATION, we need to follow these steps:

Step 1: Identify the letters available
The word EQUATION has the following letters:
E, Q, U, A, T, I, O, N

Step 2: Identify the letters needed for a 6-letter word starting with T
A 6-letter word starting with T will have the letter T in the first position, and we need to fill the remaining 5 positions.

Step 3: Count the number of ways to fill the remaining positions
Since we have already used the letter T, we cannot use it again. Therefore, we have 7 remaining letters (E, Q, U, A, I, O, N) to fill the remaining 5 positions.

To count the number of ways to fill these positions, we can use the concept of permutations. The formula for permutations is given by:
P(n, r) = n! / (n - r)!

where n is the total number of items and r is the number of items to be selected.

In this case, we need to find P(7, 5):
P(7, 5) = 7! / (7 - 5)!
= 7! / 2!
= (7 * 6 * 5 * 4 * 3 * 2 * 1) / (2 * 1)
= 7 * 6 * 5 * 4 * 3
= 5,040

Step 4: Adjust for repeating letters
In the word EQUATION, there are repeating letters:
- E appears twice
- A appears twice

To avoid overcounting, we need to divide the total number of ways by the factorial of the number of repeating letters.

In this case, we need to divide the total by 2! for the repeating letter E and 2! for the repeating letter A:
Adjusted total = 5,040 / (2! * 2!)
= 5,040 / (2 * 1 * 2 * 1)
= 5,040 / 4
= 1,260

Step 5: Multiply by the number of ways to arrange the letters T
Since the first letter of the 6-letter word is fixed as T, we don't need to consider its arrangement. Therefore, we multiply the adjusted total by the number of ways to arrange the remaining letters:
Number of ways to arrange the remaining letters = 1

Final answer = Adjusted total * Number of ways to arrange the remaining letters
= 1,260 * 1
= 1,260

Therefore, the number of 6-letter words that begin with T using the letters of the word EQUATION is 1,260.

The correct answer is option B) 2520.